Above we analytically integrated the spectral radiance over the entire spectral range. The result, Eq. (20), is the well-known Stefan-Boltzmann law. Similarly, Eq. (22) gives the integrated photon radiance. As useful as the Stefan-Boltzmann law is, for many applications a finite spectral range is needed. To facilitate this, we compute the one-sided integral of the spectral radiance. We follow the method described by Widger and Woodall[2], using units of wavenumber. Note that using other spectral units produces the same result, because it represents the same physical quantity.
\[B(\sigma)=\int_{\sigma}^{\infty} L_{\sigma}, d \sigma^{\prime}=\int_{\sigma}^{\infty} 2 \times 10^{8} h c^{2} \sigma^{\prime 3} \frac{1}{e^{100 h c \sigma^{\prime} / k T}-1} d \sigma^{\prime}\]
\[B(\sigma)=\int_{\sigma}^{\infty} 2 \times 10^{8} h c^{2} \sigma^{\prime 3} \frac{1}{e^{100 h c \sigma^{\prime} / k T}-1} d \sigma^{\prime} \quad \operatorname{let} x^{\prime}=\frac{100 h c \sigma^{\prime}}{k T}, \quad d x^{\prime}=\frac{100 h c}{k T} d \sigma^{\prime}\]
\[B(x)=\int_{x}^{\infty} 2 \times 10^{8} h c^{2}\left(\frac{k T}{100 h c}\right)^{3} \frac{x^{3}}{e^{x^{\prime}}-1} \frac{k T}{100 h c} d x^{\prime} \quad \text { where } x=\frac{100 h c \sigma}{k T}\]
\[B(x)=2 \frac{k^{4} T^{4}}{h^{3} c^{2}} \int_{x}^{\infty} \frac{x^{3}}{e^{x^{\prime}}-1} d x^{\prime}\]
Noting that \(\frac{1}{e^{x^{\prime}}-1}=\sum_{n=1}^{\infty} e^{-n x^{\prime}}\), we get \(B(x)=2 \frac{k^{4} T^{4}}{h^{3} c^{2}} \sum_{n=1}^{\infty} \int_{x}^{\infty} x^{3} e^{-n x^{\prime}} d x^{\prime}\)
The remaining integral can be integrated by parts[3]:
\[\int_{x}^{\infty} x^{\prime 3} e^{-n x} d x^{\prime}=\left(\frac{x^{3}}{n}+\frac{3 x^{2}}{n^{2}}+\frac{6 x}{n^{3}}+\frac{6}{n^{4}}\right) e^{-n x}\]
This gives
\begin{array}{rl} \label{w}\tag{23} \int_{\sigma}^{\infty} L_{\sigma^{\prime}} d \sigma^{\prime}=2 \frac{k^{4} T^{4}}{h^{3} c^{2}} \sum_{n=1}^{\infty}\left(\frac{x^{3}}{n}+\frac{3 x^{2}}{n^{2}}+\frac{6 x}{n^{3}}+\frac{6}{n^{4}}\right) e^{-n x} & \mathrm{Wm}^{-2} \mathrm{sr}^{-1} \newline \text {where } x=\frac{100 h c \sigma}{k T} \end{array}
Testing shows that carrying the summation up to \(n = min(2+20/x, 512)\) provides convergence to at least 10 digits.
Any finite range can be computed using two one-sided integrals:
\[\int_{\sigma_{1}}^{\sigma_{2}} L_{\sigma^{\prime}} d \sigma^{\prime}=B\left(\sigma_{1}\right)-B\left(\sigma_{2}\right)\]
Further, the complimentary integral is easily evaluated using (19):
\[\int_{0}^{\sigma} L_{\sigma}, d \sigma^{\prime}=\int_{0}^{\infty} L_{\sigma^{\prime}} d \sigma^{\prime}-\int_{\sigma}^{\infty} L_{\sigma^{\prime}} d \sigma^{\prime}=\frac{2 \pi^{4}}{15} \frac{k^{4}}{h^{3} c^{2}} T^{4}-B(\sigma)\]
A similar formula can be derived for the in-band photon radiance:
\begin{aligned} B^{P}(\sigma) &=\int_{\sigma}^{\infty} L_{\sigma^{\prime}}^{P} d \sigma^{\prime}=\int_{\sigma}^{\infty} 2 \times 10^{6} c \sigma^{\prime 2} \frac{1}{e^{100 h c^{\prime} / k T}-1} d \sigma^{\prime} \quad \text { let } x^{\prime}=\frac{100 h c \sigma^{\prime}}{k T}, d x^{\prime}=\frac{100 h c}{k T} d \sigma^{\prime} \newline B^{P}(x) &=\int_{x}^{\infty} 2 \times 10^{6} c\left(\frac{k T}{100 h c}\right)^{2} \frac{x^{\prime 3}}{e^{x^{\prime}}-1} \frac{k T}{100 h c} d x^{\prime} \quad \text { where } x=\frac{100 h c \sigma}{k T} \newline &=2 \frac{k^{3} T^{3}}{h^{3} c^{2}} \int_{x}^{\infty} \frac{x^{\prime 2}}{e^{x^{\prime}}-1} d x^{\prime} \end{aligned}
Noting that \(\frac{1}{e^{x^{\prime}}-1}=\sum_{n=1}^{\infty} e^{-n x^{\prime}}\), we get \(B(x)=2 \frac{k^{3} T^{3}}{h^{3} c^{2}} \sum_{n=1}^{\infty} \int_{x}^{\infty} x^{2} e^{-n x^{\prime}} d x^{\prime}\)
Integrating by parts \(\int_{x}^{\infty} x^{\prime 2} e^{-n x} d x^{\prime}=\left(\frac{x^{2}}{n}+\frac{2 x}{n^{2}}+\frac{2}{n^{3}}\right) e^{-n x}\) so
\[\int_{\sigma}^{\infty} L_{\sigma^{\prime}}^{P} d \sigma^{\prime}=2 \frac{k^{3} T^{3}}{h^{3} c^{2}} \sum_{n=1}^{\infty}\left(\frac{x^{2}}{n}+\frac{2 x}{n^{2}}+\frac{2}{n^{3}}\right) e^{-n x} \quad \text { photon } \mathrm{s}^{-1} \mathrm{m}^{-2} \mathrm{sr}^{-1} \label{x}\tag{24}\]
[2] Widger, W. K. and Woodall, M. P., Integration of the Planck blackbody radiation function, Bulletin of the Am. Meteorological Society, 57, 10, 1217-1219, Oct. 1976